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    Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand
    (Massey University, 2020) Offen, Christian
    Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed.
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    Contact systems and contact integrators : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2003) Joo, Seung-Hee
    This thesis is concerned with the study of contact systems, which are ordinary differential equations whose flow preserves a contact structure. We study contact systems from both an analytical and numerical point of view. The traditional point of view is to study the Reeb vector field of a contact form. However, if the contact Hamiltonian vanishes then its contact vector field is not the Reeb vector field of any contact form equivalent to the given one. In this thesis we study exactly this case, when the contact Hamiltonian vanishes on some submanifold of phase space. This submanifold is invariant under the flow and we study the flow on it, including its stability and fixed points. The natural numerical method for a contact system is a 'contact integrator', a map that preserves the contact structure, which is suitable for exploring the long-time dynamics of contact systems. These have not been studied very much in geometric integration. In order to formulate our results and some consequences for contact integrators, we give a thorough development of the symplectification of a contact system and have found the integrable contact systems related to integrable homogeneous Hamiltonian systems via symplectification. We develop contact integrators by the splitting method, leading to an explicit contact integrator for any polynomial contact vector field. We also study how symplectic integrators for Hamiltonian systems and volume-preserving integrators for divergence-free systems are related to contact integrators for contact systems.
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    Multisymplectic integration : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Physics at Massey University, Palmerston North, New Zealand
    (Massey University, 2007) Ryland, Brett Nicholas
    Multisymplectic integration is a relatively new addition to the field of geometric integration, which is a modern approach to the numerical integration of systems of differential equations. Multisymplectic integration is carried out by numerical integrators known as multisymplectic integrators, which preserve a discrete analogue of a multisymplectic conservation law. In recent years, it has been shown that various discretisations of a multi-Hamiltonian PDE satisfy a discrete analogue of a multisymplectic conservation law. In particular, discretisation in time and space by the popular symplectic Runge–Kutta methods has been shown to be multisymplectic. However, a multisymplectic integrator not only needs to satisfy a discrete multisymplectic conservation law, but it must also form a well-defined numerical method. One of the main questions considered in this thesis is that of when a multi-Hamiltonian PDE discretised by Runge–Kutta or partitioned Runge–Kutta methods gives rise to a well-defined multisymplectic integrator. In particular, multisymplectic integrators that are explicit are sought, since an integrator that is explicit will, in general, be well defined. The first class of discretisation methods that I consider are the popular symplectic Runge–Kutta methods. These have previously been shown to satisfy a discrete analogue of the multisymplectic conservation law. However, these previous studies typically fail to consider whether or not the system of equations resulting from such a discretisation is well defined. By considering the semi-discretisation and the full discretisation of a multi-Hamiltonian PDE by such methods, I show the following: • For Runge–Kutta (and for partitioned Runge–Kutta methods), the active variables in the spatial discretisation are the stage variables of the method, not the node variables (as is typical in the time integration of ODEs). • The equations resulting from a semi-discretisation with periodic boundary conditions are only well defined when both the number of stages in the Runge–Kutta method and the number of cells in the spatial discretisation are odd. For other types of boundary conditions, these equations are not well defined in general. • For a full discretisation, the numerical method appears to be well defined at first, but for some boundary conditions, the numerical method fails to accurately represent the PDE, while for other boundary conditions, the numerical method is highly implicit, ill-conditioned and impractical for all but the simplest of applications. An exception to this is the Preissman box scheme, whose simplicity avoids the difficulties of higher order methods. • For a multisymplectic integrator, boundary conditions are treated differently in time and in space. This breaks the symmetry between time and space that is inherent in multisymplectic geometry. The second class of discretisation methods that I consider are partitioned Runge– Kutta methods. Discretisation of a multi-Hamiltonian PDE by such methods has lead to the following two major results: 1. There is a simple set of conditions on the coefficients of a general partitioned Runge– Kutta method (which includes Runge–Kutta methods) such that a general multi- Hamiltonian PDE, discretised (either fully or partially) by such methods, satisfies a natural discrete analogue of the multisymplectic conservation law associated with that multi-Hamiltonian PDE. 2. I have defined a class of multi-Hamiltonian PDEs that, when discretised in space by a member of the Lobatto IIIA–IIIB class of partitioned Runge–Kutta methods, give rise to a system of explicit ODEs in time by means of a construction algorithm. These ODEs are well defined (since they are explicit), local, high order, multisymplectic and handle boundary conditions in a simple manner without the need for any extra requirements. Furthermore, by analysing the dispersion relation for these explicit ODEs, it is found that such spatial discretisations are stable. From these explicit ODEs in time, well-defined multisymplectic integrators can be constructed by applying an explicit discretisation in time that satisfies a fully discrete analogue of the semi-discrete multisymplectic conservation law satisfied by the ODEs. Three examples of explicit multisymplectic integrators are given for the nonlinear Schr¨odinger equation, whereby the explicit ODEs in time are discretised by the 2-stage Lobatto IIIA– IIIB, linear–nonlinear splitting and real–imaginary–nonlinear splitting methods. These are all shown to satisfy discrete analogues of the multisymplectic conservation law, however, only the discrete multisymplectic conservation laws satisfied by the first and third multisymplectic integrators are local. Since it is the stage variables that are active in a Runge–Kutta or partitioned Runge– Kutta discretisation in space of a multi-Hamiltonian PDE, the order of such a spatial discretisation is limited by the order of the stage variables. Moreover, the spatial discretisation contains an approximation of the spatial derivatives, and thus, the order of the spatial discretisation may be further limited by the order of this approximation. For the explicit ODEs resulting from an r-stage Lobatto IIIA–IIIB discretisation in space of an appropriate multi-Hamiltonian PDE, the order of this spatial discretisation is r - 1 for r = 10; this is conjectured to hold for higher values of r. For r = 3, I show that a modification to the initial conditions improves the order of this spatial discretisation. It is expected that a similar modification to the initial conditions will improve the order of such spatial discretisations for higher values of r.